1. Introduction to Numerical Methods I

2. Finite difference approximation to derivatives

3. Consider a smooth function g(x). Taylor’s theorem reads:
In particular:
Short course on: Numerical methods for hyperbolic equations and applications-Trento, Italia-June 7th to 18th, 2004

4. Finite difference approximation of PDEs: FTCS
Discretise the x-t domain into a finite number of M+1 points, I=0,…,M

5. Now approximate partial derivatives.
Use finite differences:
Then the PDE is replaced by a finite difference approximate operator:

6. Introduce the dimensionless number:
The Courant-Friedrichs-Lewy number
or CFL number, or Courant number
Note that c is a dimensionless quantity, it is the ratio of two velocities:
Finally, the FTCS scheme reads:
Stencil
This formula allows us to calculate explicitly the evolution in time of discrete approximate values of the solution at every point, except for i=0 and i=M.

7. Convergence
Our ultimate goal is to construct schemes that converge, that is schemes
that approach the true solution of the PDE when the mesh size tends to zero.
Lax’s Equivalence theorem says that:
the only schemes that are CONVERGENT are those that
are CONSISTENT and STABLE
We must therefore work on CONSISTENCY AND STABILITY.

8. Local truncation error
Consider a particular scheme: FTCS.
The numerical analogue of the PDE is the approximate operator:
We define the local truncation error:

9. The FTCS is a first-order scheme
Assuming the solution is sufficiently smooth we Taylor expand and obtain:
Noting that:
The FTCS is a first-order scheme
In general, if the local truncation error of a scheme is of the form:
Then the scheme is said to be k-th order accurate in time and l-th order accurate in space.

10. Consistency For example, for the FTCS scheme we have
A numerical scheme is said to be consistent with the PDE being discretized
if the local truncation error tends to zero as the mesh size tends to zero.
For example, for the FTCS scheme we have
Therefore FTCS is consistent wit the PDE

11. Stability of a numerical method.
If a method is consistent with the PDE, then all we need to bother is stability.
One view of stability is that of unbounded growth of errors as the numerical
scheme evolves the solution in time.
Another view of stability is that of controlling spurious oscillations
Stability in the sense on unbounded growth can be analysed by a variety of methods
A popular method is the von Neumann method
One performs a Fourier decomposition of the error. It is sufficient to consider a single component.

12. Stability analysis of the FTCS
Thus FTCS is unstable under all circumstances: UNCONDITIONALLY UNSTABLE (useless).

13. Godunov’s first-order upwind scheme
Approximate derivatives in by
then
The scheme reads
Illustrate the
stencil

14. Local truncation error:
The finite difference operator is:
Substitution of the exact solution into the approximate opetaror gives:
Assuming sufficient smoothness and Taylor expanding:
The scheme is first-order accurate in space and time

15. and the stability condition becomes
Stability analysis
and the stability condition becomes
The CFL condition of Courant condition
Given wave speed and mesh spacing the time step is determined from the stability condition

16. The stencil (upwind) t
n+1 o n o o x
True domain of dependency
Numerical domain of dependency
The exact solution is the value on the characteristic
where is the foot of the characteristic at time

17. For appropriate choices of the point lies between and
Assume a linear interpolation between and
Evaluation of
gives
which is the Godunov scheme

18. The “downwind” scheme Approximate derivatives in by then
The scheme reads
Exercise: derive the local truncation error. Is the scheme consistent ?
Exercise: show that this scheme is unconditionally unstable.

19. General Form of the First-Order Upwind Scheme
For the upwind scheme is
For both and
define:
The scheme reads:
Exercise: show that the upwind scheme for negative speed is conditionally stable

20. Fully discrete and semi-discrete schemes
If the time derivative is left in its continuous form
Semi-discrete
The method of lines

21. Explicit scheme and implicit schemes
Exercise: construct the fully discrete implicit version of FTCS

22. Monotone Schemes A monotone scheme satisfies:
Monotone schemes for the linear advection equation with constant
speed of propagation are those whose coefficients are non-negative.
Example: The Godunov upwind scheme.

23. More Schemes Conditionally stable Monotone Modified equation has
The Lax-Friedrichs scheme (LF)
Conditionally stable
Monotone
Modified equation has
LF may also be seen as the FTCS scheme (unstable) with replaced by
Note the shape of the stencil.

24. The Godunov Centred Scheme
Stencil
Conditionally stable
Monotone for
Oscillatory for
This is an interesting example of a first-order scheme that is NOT MONOTONE.

25. The Lax-Wendroff Scheme (LW)
Stencil
Conditionally stable
Non-monotone (verify by inspection)

26. Conditionally stable:
The FORCE scheme
stencil
Conditionally stable:
Monotone
Modified equation:
© Toro 2004